Mathematical Physics · arXiv:0707.4143 v2 · submitted 27 Jul 2007
Hopf algebra of non-commutative field theory
The paper says
“We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.”
from the abstract · picked out as the main claim by a classifier
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The paper
Abstract
We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory.
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abstract compiled · 2026-09-23 · no language-model review
How the main claim is stated: asserted 1.00
What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.04 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.03 · mentions code or data availability 0.02
The abstract, sentence by sentence
This table labels the abstract only (arXiv listing abstract v2). The number is the probability the classifier gave the label.
| Sentence | Role | The authors' own work | States the main conclusion | Strength |
|---|---|---|---|---|
| We contruct here the Hopf algebra structure underlying the process of renormalization of non-commutative quantum field theory. | contribution 1.00 | 0.95 | 0.68 | asserted 1.00 |
What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00
What was examined, and what was not
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- arXiv listing abstract v2
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- full text
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- P abstract
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- S sections
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- R cross-paper relations
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 1,592 input tokens, $0.0001 reported.
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Cited by 1 Pith paper
Reviewed papers on Pith that cite this one. The full citation record
- A translation-invariant renormalizable non-commutative scalar model math-ph · Feb 2008
How the literature uses it
25 arXiv papers cite this in 36 sentences
Papers citing it · 24 sentences without a confident label
Builds on it 1 paper
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The proof of Lemma $3.2$ of [ref] generalizes in a straightforward manner.
Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory · The assigned graph combinatorial Hopf algebra · 0.90
Uses its method or tool 1 paper
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Given the above Lemma, we may now follow along the lines of the coassociativity proof for the coproduct in [ref] .
Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory · Hopf algebraic description of the combinatorics of the renormalizability of the BGR model · 0.85
Compares with it 1 paper
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However, they allow the possibility to insert graphs also in different ways but show that there exist the right (cyclic) insertion to prove coassociativity [ref] .
Combinatorial Dyson-Schwinger Equations of Quartic Matrix Field Theory · Hopf algebra of perturbative renormalization · 0.88
Cites it as background 7 papers
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In quantum field theory (QFT), a similar success is the elegant description of the combinatorial backbone of perturbative renormalization via the combinatorial Connes-Kreimer Hopf algebra on Feynman graphs (see the original paper [ref] as well as section $1.6$ of the book [ref] )[Let us also mention here that a Hopf algebraic description was also used to describe the combinatorics of perturbative renormalization on noncommutative Moyal space scalar QFT (where graphs are replaced by ribbon graphs, or combinatorial maps) [ref] , [ref] ; moreover, Connes-Kreimer-like Hopf algebras have been defined for quantum gravity spin-foam models [ref] , [ref] .].
Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory · Introduction and motivation · 0.81
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Connes-Kreimer-like Hopf algebras have been defined for some renormalizable Moyal [ref] , [ref] and sixth order interaction tensor models [ref] .
Renormalization and Hopf Algebraic Structure of the 5-Dimensional Quartic Tensor Field Theory · Introduction · 0.88
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For recent approaches in the framework of noncommutative field theory, one can consult [ref] ).
Parametric Representation of Rank d Tensorial Group Field Theory: Abelian Models with Kinetic Term $\sum_{s}|p_s| + \mu$ · Factorization of the amplitudes · 0.98
What it cites · 22 sentences without a confident label
Uses the method or tool of 1 paper
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The parametric representation constructed in [ref] writes the Feynman amplitude $ϕ(Γ)$ as [equation] where $K$ is some constant, [equation] where $α_ℓ$ are the parameters associated to any of the propagators of the graph.
Parametric Representation of Noncommutative Field Theory · Renormalization as a factorization issue · 0.90
Cites as background 8 papers
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We will mainly focus on the Grosse-Wulkenhaar model [ref] or non-commutative $Φ^4_4$ theory.
Renormalisation of \phi^4-theory on noncommutative R^4 in the matrix base · Renormalization of non-commutative quantum field theory · 0.96
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We will mainly focus on the Grosse-Wulkenhaar model [ref] or non-commutative $Φ^4_4$ theory.
Renormalization of Non-Commutative Phi^4_4 Field Theory in x Space · Renormalization of non-commutative quantum field theory · 0.96
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Note that even though recent progress has been made in [ref] , physicists do not yet have a renormalizable non-commutative gauge theory.
Non-Commutative U(1) Gauge Theory on R**4 with Oscillator Term and BRST Symmetry · Introduction and motivation · 0.86
Challenges and discussion
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